Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere

TL;DR

This paper studies how rational points distribute on the sphere, improving bounds on their equidistribution in small regions and exploring connections to Diophantine approximation and classical number theory conjectures.

Contribution

It provides improved bounds for small-scale equidistribution of rational points on the 2-sphere using Hecke operators and links these results to covering radius and Diophantine approximation problems.

Findings

Enhanced small-scale equidistribution bounds for rational points on the 2-sphere

Connections established between equidistribution, covering radius, and Diophantine approximation

Discussion of implications for Linnik's conjecture on sums of two squares

Abstract

We consider rational points on the sphere and investigate their equidistribution in shrinking spherical caps. For the two-dimensional sphere, we leverage Hecke operators to obtain a significantly improved small-scale equidistribution bound, and discuss connections to the covering radius problem, intrinsic Diophantine approximation, and Linnik's conjecture on sums of two squares and a mini-square.